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ZILBER's PSEUDO-EXPONENTIAL FIELDS the Goal Of ZILBER'S PSEUDO-EXPONENTIAL FIELDS WILL BONEY The goal of this talk is to prove the following facts above Zilber's pseudoexpo- nential fields (1) they are axiomatized in L!1;!(Q) and this logic is essential. (2) they form a quasiminimal class ∼ (3) C ≡ B implies C = B The results and presentation are drawn from Kirby \A note on the axioms for Zilber's Pseudo-Exponential Fields" [Kir] and Bays and Kirby \Excellence and uncountable categoricity of Zilber's Exponential Fields" [BaKi]. ∗ The class of pseudo-exponential fields is denoted ECst;ccp and is the collection of structures hF ; +; ·; expi satisfying the following axioms: · (1) ELA-field: F models ACF0 and exp : F ! F is a surjective homomor- phism (2) Standard kernel: ker exp is an infinite cyclic group generated by a tran- scendental element 2πi. (3) Schanuel property: For all x 2 F , δ(x) := td(x; exp(x)) − ldimQ(x) ≥ 0 (4) Strong exponential-algebraic closedness: Every system of exponential polynomials has a solution, unless this would violate Schanuel. (5) Countable Closure Property: eclF is a pregeometry such that, if C ⊂ F is finite, then eclF (C) is countable. The fourth axiom is imprecisely stated, but we will state it more precisely. We will change the language (essentially but conservatively) to show this is a quasiminimal class. First, we define some useful notions. δ(x=A) := td(x; exp x; A; exp A=A; exp A) − ldimQ(x=A) Second, we discuss the axioms. (1) ELA-field: Clear. Date: August 12, 2015. This material is based upon work done while the first author was supported by the National Science Foundation under Grant No. DMS-1402191. 1 2 WILL BONEY (2) Standard kernel: This is L!1;! expressible. The important bit is that this defines the integers: Z(F ) := fr 2 F : 8x[x 2 ker exp ! rx 2 ker exp]g An alternate approach would to just say that Z(F ) is standard (using L!1;!); then the standard kernel is first-order. (3) Schanuel property: This is expressible as a first order scheme, modulo n n the integers being standard: for each variety V ⊂ Ga × Gm defined over Q of dimension n − 1, n ! X 8x9m 2 Z(F ) − f0g (x; exp(x)) 2 V ! mixi = 0 i=1 Essentially, this says that any tuple with td(x; exp(x)) < n has some linear dependence. This property is a big part of the interest in Zilber's work. Schanuel's Conjecture says that C has satisfies this and, not to understate it's im- portance, Wikipedia1 says something like proving Schanuel's Conjecture \would generalize most known results in transcendental number theory." (4) Strong exponential-algebraic closedness: Before we state this for- mally, we need some definitions. Write G for Ga × Gm. n Given a matrix M 2 Matn×n(Z), this acts on G by being an additive n n map on Ga and a multiplicative map on Gm. Write G · V for the image of V ⊂ Gn under M. An irreducible variety V ⊂ Gn is rotund iff for every M 2 Matn×n(Z), dimM · V ≥ rkM. Let (x; y) be a generic point of V over F . V is additively free iff the xi don't satisfy X mixi 2 F i<n for any mi 2 Z, not all 0. Similarly, V is multiplicatively free iff the yi don't satisfy mi Πi<nyi 2 F for any mi 2 Z, not all 0. The property is: n n If V is a rotund, additively and multiplicatively free subvariety of Ga ×Gm defined over F and of dimension n and a is a finite tuple from F , then there is x 2 F such that (x; exp x) 2 V is generic in V over a. So if there are some exponential polynomials that are nice enough, then there is a root. 1http://en.wikipedia.org/wiki/Schanuel's_conjecture#Consequences ZILBER'S PSEUDO-EXPONENTIAL FIELDS 3 Proposition 0.1 ( [Kir]). This is first-order expressible modulo the previ- ous axioms. Proof: The key is that we have Z already definable. Suppose we have n a parametric family of varieties (Vp)p2P from G , where P is the parame- terizing variety. It is known (fibre dimension theory) that fp 2 P : Vp is irreducible of dimension ng is first order definable. 2 With Z , we can define additive freeness: ! X 8m 2 Z − f0g8z9x x 2 Vp ^ mixi 6= z i<n This gives additive freeness: the generic over F satisfies all equations over F that every member of Vp does, and this says that, for each element of F , there is some tuple that doesn't linearly combine to it. Similarly, one can get rotundness and multiplicative freeness3. Alterna- tively, we can appeal to [Zil05, Theorem 3.2] says that these are first-order definable in the field language. n Given a parametric family of varieties (Vp)p2P from G , where P is the 0 parameterizing variety, let P ⊂ P be the subset so Vp is irreducible, of dimension n, rotund, and additively and multiplicatively free. Then we add the following scheme 0 r n 8p 2 P 8a 2 F 9x 2 F 8m 2 Q " !# X X (x; exp x) 2 Vp ^ mixi + mn+iai = 0 ! ^i<nmi = 0 i<n i<r In a sense, this says that spanfxg \ spanfag = f0g and x is linearly inde- pendent. First, note that this follows from axiom 4: Given such a Vp, there is (x; exp x) 2 Vp which is generic over a. By additive freeness, this means that x does not Q-linearly combine to an element of F , which this gives. Second, suppose this holds. Let V be a rotund, additively and multiplica- n n tively free subvariety of Ga × Gm defined over F and of dimension n. Then 4 it is Vp in some parametric family . Let a 2 F . Adding things to a, we may assume that δ(y=a) ≥ 0 for all y 2 F 5 and that V is defined over a. 2 Necessary; x1 + px2 = 0 is additively free iff p 62 Q 3Note that xy is definable as exp(y log x), which is well-defined for integer y. 4Parametric varieties are the equivalent of fixing the degree and quantifying over all coefficients in algebraically closed fields. 5To do this, let a0 be a minimizer of δ(y=a) as y ranges over F . Then we have δ(y=aa0) = δ(yaa0) − δ(aa0) ≥ 0 (by choice of a0), so use aa0 in place of a. 4 WILL BONEY The scheme then gives a x such that ldimQ(x=a) = n. By Schanuel, then td(x; exp x=a; exp a) ≥ n. Since V is of dimension n, we have (x; exp x) is generic in V over (a; exp a), which is more than we need. y [Kir] says that this was originally described by [Zil05] as a \slight sat- uratedness" of the structure, so it being first order was a bit of a surprise. Having it first order allows for the some of the details to be simplified. On the other hand, the analogy to algebraic closure makes it being first order more expected. (5) Countable Closure Property: There are two definitions of exponential closure that are the same with axiom 2 (we will not prove this, see [Kir10, Theorem 1.3]). We define the one used in [Kir]. An exponential polynomial f(X) = p(X; exp(X)), where p 2 F [X; Y]. A Khovanskii system of width n is exponential polynomials f0; : : : ; fn−1 of arity n such that fi(x) = 08i < n @f0 @f0 ::: @x0 @xn−1 . .. (x) 6= 0 @fn−1 ::: @fn−1 @x0 @xn−1 Then F ecl (C) := fa0 2 F : a 2 F is the solution to the Khovanski system given by f0; : : : ; fn−1 defined over Q(C)g By [Kir10, Theorem 1.1], this is a pregeometry in any exponential field. We also have an easy proof that this is L(Q). Proposition 0.2 ( [Kir]). Axiom 5 is an L(Q) scheme. Proof: Given exponential polynomials , let χ(x; z) state that x is a solution to the Khovanskii system with coefficients z. 8z:Qx09x1; : : : ; xn−1χf (x; z) y We're going to discuss two interesting things coming from [Kir] and then look at the language used for quasiminimality. Q cannot be eliminated: We're going a little out of order here, because we will use some of the quasiminimality in this proof. Proposition 0.3 ( [Kir]). Countable closure is not L1;! expressible. ZILBER'S PSEUDO-EXPONENTIAL FIELDS 5 Proof: Let F0 be the zero dimensional Zilber field. Then we can adjoin some a 6 so exp a = a and form the strong exponential algebraic closure to get F1 ∼ . Note that F1 is also a zero dimensional Zilber field, so F0 = F1. Our goal is to show that F0 ≺L1;! F1. Let b 2 F0 and let B be the smallest ELA-subfield containg b. By general Fraisse construction or [Kir11, Proposition 6.9], both F0 and F1 are isomorphic to ∼ the strong exponential algebraic closure of b. That is F0 =B F1. Thus, given any L1;!(b) formula, F0 and F1 agree on it. We iterate this process to form fFα : α < !1g. Note at limit stages, L1;! substructure is well-behaved (L(Q) is not) and taking unions cannot grow the ex- ponential transcendence degree. Thus F!1 is L1;! equivalent to all of these, but does not satisfy the countable closure property.
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